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COSC 6373: Computer Vision - Lecture15
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00:02 all right. Good morning. Um today we gonna do on started discussion
00:11 new topic girl, which is So, um, so far,
00:17 been talking about, um, designing Victor's on in general, trying to
00:25 how we build an ability to recognise , certainly in the context of
00:33 but also the context of trying to some of the objects of similar
00:39 So one of the other, open problems that has a very broad
00:47 of applications is tracking in tracking. could be relevant to, you
00:53 essentially detection. And, um, at how a particular object might,
01:02 know, move in space and time given observation. So tracking has multiple
01:11 . It's essentially trying to locate Ah, that potentially of interest to
01:17 . Of course, identifying what those might be can be part of the
01:21 problem. But the other aspect is actually deter mined the dynamic configuration,
01:28 is how the object is actually moving changing positions over time. Eyes critical
01:36 addressing the tracking problem. In a , it's ah, the glint of
01:41 Teoh follow something with their eyes. . That object goes very able to
01:48 it and track it. Ah, off all the challenges that might be
01:58 in the context of seems realization when the objects might actually appear really partially
02:05 the scene or the might be similar other objects. Eso can be
02:11 toe, identify and recognise a particular and so on. So far,
02:16 we'll look at each one of these to talk a little bit more about
02:20 problem. So here's an example off tracking, and what you see is
02:27 a bounding contour around the object appear tracking. In this case, a
02:34 , says the person who was in scene, our corn to responded and
02:39 able to localize the sort of the toe, the outline off the
02:47 And in a way, you can of this problem as saying that you
02:51 , we are tracking the center object object being start off as one
02:58 or we can also think of this you say, Well, I'm really
03:03 tracking the boundary. The boundary is up of multiple points, and each
03:09 is something that I'm actually being able match, that it's Ah, changing
03:16 over time, right? So, , a a very abstract sense.
03:24 is really about looking at a position evolving its position with tiny right.
03:32 , as I said, there are broad range of applications for tracking.
03:38 can be thought of as a low Albert of them, if you
03:41 for many other applications. Things like computer interaction. Right. So these
03:48 , um, you know, you our that was, in the context
03:54 augmented reality. You're essentially embedding real into a virtual world, if you
04:01 , in order to I understand how real object is exchanging and essentially be
04:07 to model that in the virtual We need to be able to track
04:11 geologic on DNA. In most the tracking is happening with respect to
04:18 riel objects in the scene. great example of this, you
04:23 that came about a few years back when Microsoft introduced the connect,
04:30 So the connector has two types of . If you know it's actually got
04:35 infrared sensor along with an infrared great on then, of course, got
04:42 visual sense in Israel. A visual on what it allows want to
04:48 Is this to essentially track of person based on their movements? It's able
04:59 sort of allow for interaction in a world. A game, if you
05:07 . It would be one example. was ah, beyond human computer interaction
05:13 reality and so on. Ah, applications and security and surveillance, of
05:19 . Ah, you know, because you can imagine in public safety
05:23 , there is a need to try understand how objects move potentially identify
05:30 collide with each other things like, know, traffic accidents and so
05:33 so forth. Ah, and at same time, there might be applications
05:39 relate to security Being able to, , you know, clack a
05:45 for example. You know, if were to, um, you
05:51 get into a public fight, you have to be able to tease out
05:56 are the people involved where they move maybe before or after a particular
06:03 People Teoh identify directions in which there may have traversed, store or
06:09 and so on so forth. So are broad range of applications for
06:15 um, in already off scenarios now day the court affect right? I
06:24 , we have to understand, from computer vision standpoint, what is striking
06:29 was really supposed to do, So in a very simple sense,
06:35 idea is that, given an input , right, if you have a
06:39 , uh, let's say for sea this, people walking down tracking has
06:46 essentially be able to tell us there the objects of interest in this case
06:53 , uh, at the cutter And, of course, it has
06:57 be able to tell us what is particular trajectory off those objects prior to
07:06 current time. Right? So it's important that a tracking elegant of them
07:12 its core be able to deliver two of information, which is the current
07:17 for objects of interest and how they to that current position over time.
07:24 ? So that's that's really what we're Teoh build or design in an Alberta
07:30 when we talk about a tracking are tracking out. So let's look
07:35 this problem on break it down into the elements Ah, that may be
07:43 for us to think about when we about a tracking Al, right?
07:47 let's look at a cell for, know, sort of Ah,
07:52 Ah scenario. Let's say you you have a whole way the person
07:59 you're interested in tracking and that's the person in the scene for now.
08:03 , so first and foremost, of , we want to be able to
08:08 the object of interest. Now, know, we've been talking about
08:13 detectors for the pastor. You I guess a few lectures eso you
08:20 how to try to go ahead and a detector. In this context,
08:24 talking about a person detectors so we do that. And we know the
08:28 will give us a bounding box around person. Now, of course,
08:31 know, we want to be able think about tracking where tracking tells is
08:37 position of the object we're interested So you know we can You can
08:42 about a position in two D. can pick any any position. Let's
08:47 we pick the position that the, know, on the lower edge of
08:53 bounding box, maybe the center one pick any other position could be the
08:57 right off this object. A but the central of the Barney box
09:02 be picked as the position. But in this example use this particular
09:06 the one as the position of the product. So now it At some
09:13 after that position, the object will mood and will appear in a different
09:19 . Now we can basically do our again on the particular image, and
09:24 get abounding walks. And similarly, know, we find the center off
09:29 lower edge of the Barney marks and us a new position. Let's say
09:33 T So piece of tea is our position. Great. So now we're
09:38 to actually just apply our detector over over again to be able to identify
09:45 object that we want to localize and detection. It gives us the position
09:51 the object. Ah, and of , Ah, Now, if I
09:56 all dispositions in time, I will up getting a trajectory. So in
10:01 baby, my tracking problem is solved it gave me the position of the
10:08 in every image. Give me on to connect all dispositions and essentially come
10:15 with. Ah, a temporal evolution points, Which tells me how that
10:21 object came to its current position in present time. Right now. In
10:26 case, of course. Ah, know, detection is what I'm
10:30 So we can call the striking by kind. Of course, all we
10:34 have is the bottle that we used while building our detector. Ah,
10:42 let's say you know, we be , for example, the shape you
10:46 , we could have used, uh , everything that you talked about so
10:51 , right? Hard descriptors are hard . Any one of those is what
10:57 could have used in our design, detector. But at the end,
11:01 end up being able to do uh, tracking on, in this
11:06 , tracking bad detection because that's all doing now, of course, as
11:11 know, um, you know, are not perfect. One of the
11:16 that we, uh, have talked is the fact that, you
11:20 detectors can produce false detections. So say Ah, for whatever reason,
11:29 , we come back to our simple . Single person in the scene.
11:34 doing our detection. We detected the at disposition Piece of wanted at the
11:40 , we have the person moving. at some point in time, let's
11:46 time, Step K. We got false detection from our detector, and
11:51 ended up finding this particular bounding box on, Of course, because we
11:56 a bounding box, we assume we you know, since you're not see
12:01 visualize ing every detection, we assume the defections are correct. We get
12:05 detection and of course, we get location of the bounding box.
12:10 the consistent way that he chose and call it Piece of cake.
12:15 of course, if I, you know, I'm simply creating the
12:19 by connecting all these points. The Lee I will get is sort off
12:24 I'm showing here. Um, so have, ah, evolution of the
12:29 with all these points were potentially one the points is not quite correct.
12:36 this can be problematic, because now don't necessarily have the correct trajectory re
12:41 . There are trajectory. The other that detectors is of course, you
12:47 , if we are unable to actually the object very well for various
12:53 One of them, ah, being obvious one, which is the object
12:57 occluded. So in this example, say there are objects in front and
13:00 can't quite see the object of interest person in our detection completing fields.
13:07 if that happens like that at a time point, we will simply not
13:12 any detection. Um, so both these issues related to detectors can kind
13:22 be problematic, right? One is detections. The others. Ah,
13:27 . And in either case, we not necessarily be able to get a
13:32 trajectory, and as a result, tracking solution is not quite to the
13:40 that be made desire. Now, course, there are other problems with
13:44 object detections, you know, cause , for example, of you talked
13:49 this rare once we train a Typically our training is is made up
13:56 example er images. And if exemplar don't capture all the potential variations that
14:02 might encounter Ah, during testing our detector may not quite toe work
14:08 well, right? And of when we're talking about moving objects,
14:12 gonna be a lot of aviation's posed elimination radiations could be Could be two
14:19 types. There could be others. in either case Ah, you
14:22 detection may or may not be quite par. And as that translates
14:28 ah, the accuracy of tracking as . Now the other thing is,
14:35 course, when we have multiple objects the seat, right. So let's
14:39 instead of having one person that be tracking now we have two people that
14:44 tracking. Okay, we can still our detection. And of course,
14:49 we do that, we'll end up two detections. Right? So we'll
14:52 to bounding boxes because they're two people the scene and of course,
14:57 we have at some point in as the Two Objects movie have grasped
15:04 detection belongs to which object, So rich detection would belong to which
15:10 in this case. So the idea piecing this out when we have multiple
15:16 in the scene and making sure that associate the correct detection at each time
15:23 . So the correct object that be tracking This problem is called data
15:28 right? So we have to know , rich observations are our measurements were
15:34 go with which particular I d often that we're tracking. Now, this
15:41 be quite uninterested. Problem, And typically data association. One way
15:49 think about us that maybe if I , um you know, I kind
15:54 know a little bit more about the that I'm tracking. Uh, maybe
15:59 additional information can help me solve the association problem. Right. So let's
16:05 a simplest strategy that they can apply that maybe as we get every new
16:11 , remember, For every new we are computing the current position off
16:17 particular object. So one simple we're did association is to say that why
16:23 assigned the current detection? We have the i d for which the location
16:34 the current detection is as close as to the location off the previous
16:40 Right. So this would be a simple strategy of saying, choose the
16:45 that is close us to the last . Okay, So if for
16:52 I have ah, detection, you know, let's say you
16:59 I'm tracking the red dot right? I end up seeing two detections.
17:05 of these purple dots. The question , which one is the red I
17:12 And you can say Well, you , let me look at the distance
17:16 the red and this purple dark and red and this purple dark and
17:21 this seems to be the closest the are that this is a purple
17:25 Exactly the same has the red dot d and I make those two association's
17:32 so that that's the idea. So be essentially end up doing that.
17:37 say they have the two objects, know, we find their locations that
17:42 detect and the object booze. You , we'll end up finding to new
17:47 boxes and, of course, the location of the object. And I
17:51 , Well, you know, obviously I look at this this blue dot
17:56 this late daughter closer to each other to this blue in this lead and
18:00 same for this So you know, two should be associated together belonging to
18:06 same object. Great on that kind works off course. Ah ah,
18:13 to think about that, you in the formulation is that you
18:16 if I have UH, two P and Q. Okay, And
18:22 course, I am creating the trajectories time. So p one teau p
18:27 T minus one is the trajectory of . P. Q. One to
18:32 T minus Juan is the trajectory of . Q. Sorry, this should
18:36 cute. And then I get to directions in one attempt to on.
18:41 gonna do the Data association for computing of T and Q sub team to
18:48 equal to the minimum. Off the off P of T minus one
18:56 Am I plus the distance off Q minus one MJ, right. So
19:03 I. For all all possible choices . Am I and M J that
19:07 have. I have to compute this off distances and the one that is
19:13 smallest should be the correct value for minus one. Q T off minus
19:20 in order to get the next Okay, that's the idea. Of
19:25 , this will only hold true if do the closest distance association now off
19:31 , This kind of works well. , you know, be generally do
19:36 know what we're not accounted for. the objects are moving. So what
19:41 if the two objects that were trying detect tend to sort of say that
19:47 moving essentially in a way that they come closer to each other?
19:53 So? So if both of them taboo sort of towards the center location
19:57 to the two of them, we end up saying getting to detections at
20:02 next time point that are very close each other. Now this, then
20:05 comes lock tougher, right? um, the distances Ah, for
20:12 the previous position. And the next can become very, very similar.
20:19 . Off which association be, you , sort of which radar be associate
20:26 the rich blue dot Right. Eso may not give us a good
20:31 So, um, this strategy off the closest location may not necessarily translate
20:40 in all applications. No, imagine we had some more information that we
20:48 leverage. Right. So as we're the two objects, the two people
20:53 this case let's say that one of objects is actually bearing, or the
21:00 of the object is different between the objects that we're looking at,
21:05 So in the simple example, you , given, given this this person
21:11 , colored shared compared to this person , as we saw try to solve
21:17 state association problem. What if we were to look at our detection results
21:23 say that well, rather than just the bounding box with one indicator,
21:29 is the position of the burning How about during the later association?
21:35 also use the information that we have the object within the bounding box?
21:41 ? So one example would be to , Why not compute the color
21:45 Graham. Let's say off the pixels the bounding box and use color history
21:52 as a representation off the appearance off object. Great. Now, if
21:57 news that, then we can say my deed association problem can we now
22:04 as not only solving for association by the closest distance off the previous object
22:15 to the new object position, but by maximizing good match off the color
22:24 off pixels within the bounding box off detected object. So let's say the
22:31 color history grams for each of P each of Q R color Instagrams for
22:37 two objects. P and Q. of course, what I want to
22:40 is to match not only the distance , you know, between Ah PT
22:49 one and the new detection. Am and Q T minus one in the
22:53 MJ. But I also want to the distance. Uh, that I'm
23:01 for the history Graham off p instagram am I blessed instagram of Q on
23:09 A for him, Jay. So we want to make sure that
23:11 similarity between the history Grams doing data or maximum strikes and we have one
23:17 the stroke itself information now that being so are me an association based on
23:24 Israelis color match. No, this be quite interesting. Of course,
23:31 still left with the issue off a and false alarms when we're during
23:36 Right? So we talked about this , so I'm a potentially get a
23:40 production, and I may actually not able to observe one of the objects
23:45 one particular friend. Great. I mean the skin stilyan issue,
23:51 we have some robustness in the sense on doing data association and it is
23:57 step to make sure that our tracking you know, more robust algorithms right
24:07 . Of course. Um, what kind of information can be news than
24:13 attracting objects? Well, since we're objects, you actually know Stempel
24:20 The different positions the object has taken time. Now there's the object is
24:27 moving in space. Potentially, we ability to compute the velocity at which
24:35 object is moving. So if I that the object is is it a
24:41 location at let's 18 subzero then of , the object Each of them will
24:48 some velocity with which they will move as a result, potentially come to
24:54 new position at a next time. right. So the velocity, let's
24:59 it. But with the vector V now, of course, if I
25:04 that Vector V, then I have ability to kind of know where the
25:10 will be at a time except say right, going from 0 to
25:15 And the way I can know that estimate that is simply by solving,
25:21 , the simple equation where we know velocity is just a change in position
25:27 time Eso I can lead. I say it will be one which is
25:32 position of time 0.1 building close to position of zero by position at
25:40 You know, plus t one minus 0 13 multiplied by the velocity.
25:47 so this gives me an ability to of know bear, the object will
25:52 at time pointy one. Well, is great. Now, if I
25:57 this, then I don't actually have search my end that image for potentially
26:05 the object is. It actually gives a more bounded search space to
26:11 Hey, look for the object. , Within this area off the image
26:17 said this is helpful because it will help reduce the false detections that I
26:24 be getting from my detect. So a more confined search space can help
26:30 eliminate false protections and as a of course, held and improve our
26:36 accuracy. Do we get better were able to do better data
26:41 and as a result, hopefully do better job of track in a
26:46 As I do this, I can updating not only the position of the
26:53 . But now, if I know where the object is by doing a
26:58 job of detective addiction, I can use the new location to go back
27:04 update my velocity information about the So I have a better idea on
27:10 to predict its next location. So we have a dig, a strategy
27:17 data association strategy that potentially is becoming and more powerful, which has been
27:24 only just doing association but proximity, doing association by appearance rate. So
27:30 a matching colors could be other features well that we can use, but
27:35 way off matching information that we extract the pixels within the bounding box of
27:41 detective object. And then, of , Ah, you know. Now
27:45 are also saying that they are going improve our did association by using
27:53 right? So by not only the point match, but using the prediction
28:01 saying what is the closest detection to predicted location? Are there that object
28:10 expected off beer based on the velocity that we have about the art?
28:16 , so this now becomes quite powerful in fact, using this they are
28:22 did association. We may even be to deal with partial exclusions or temporary
28:34 Zoff objects. So let's say we unable to detect an object at one
28:39 time point. But I can still able to predict potentially where that object
28:47 appear at the next time point. my velocity information, I may be
28:54 to continue doing prediction. Using the known velocity and previously known position often
29:03 even in the case when an object to be detected by the detector at
29:09 particular time point. So this becomes even more robots straight. So here's
29:15 example. Right again, we're using quite simple. We are looking at
29:22 color information off this particular ball in example. But if you think about
29:28 , we are able to actually track continue tracking the ball, even burn
29:33 ball actually goes outside off the frame observation. Right. So this is
29:42 is a simple example of how we do tracking by detection, uh,
29:47 some simple strategies. Okay, no, if you think about
29:51 are tracking algorithm as you ward We are using one sort off the
29:58 detection, right. Some kind of shape information over the object of
30:03 We're using the appearance. That's a information or color history Graham off the
30:08 within the bounding box. And then also using the dynamics whore some motion
30:16 about the object in order to continue with objects successfully. Right? So
30:23 using multiple types of information in order solar tracking problem. Okay, so
30:30 , tracking Bye bye. Um, this information gives us a much more
30:36 a solution to doing tracking. of course, keeping moment. Some
30:42 this is based on prediction rates are is based on our understanding off the
30:49 of an object. And prediction um is our model and using detection
30:58 is a really for the still verified our model is telling us by essentially
31:07 what is happening in the space. we're gonna be had these two objects
31:11 the scene. The objects are at particular location. A time 0.0.
31:17 . So that's it is blue Of course, we have our
31:21 And based on the velocity, we predict there, that object will be
31:25 time 0.1, but at the same , given an image that we get
31:32 time 0.1, we have our detector our detector gives us an estimate of
31:37 bounding box off where the object And given that bounding box, we
31:42 a better estimate of their exactly that is so the yellow dots signify the
31:48 position. Right now, once we all three, we actually also have
31:55 ability to know potentially what are detection is or what our prediction. That's
32:02 . And that's given by the difference the green dark and the yellow
32:07 So the difference in the prediction and is important. Tees off information that
32:12 need to account for is we kind get ready to apply are tracking framework
32:19 the next time point. So we to do some correction when we are
32:25 using dynamics for tracking. So here's base case. Let's assume that we
32:31 some initial prior that predicts that state the absence of any of its rights
32:35 your no observation as yet, But course we are inst enshi ating,
32:42 you will, a couple variables that us information about the object now?
32:47 course. What are those variables? , in this simple example, we're
32:51 about the position in the velocity off object that we're interested in tracking
32:57 Tell us what? Is the state the object? No. When we
33:03 the first frame, we have our observation, right? And the minute
33:08 ever observation, we don't our We actually get the correct opposition off
33:14 objects were given. Ah, some . And for ritually actually have what
33:20 consider to be the correct value. we given the correct value and give
33:26 our ability to predict that value gives now an ability to correct our estimate
33:35 the given time point. Right? given something that we actually have
33:40 we can now use that information to something in the future. Off course
33:46 in the future, when the future present, right? So we have
33:49 observation. A time 10 point, one. They can now have the
33:54 to correct our information and then predict time point D plus two. And
33:59 keep repeating this process forward and over now. So let's come back to
34:07 looking at this and thinking about this a very, very simple sort of
34:11 model system that said, We have ball that were instruments train tracking the
34:17 . Is that position eg zero. we said that, you know,
34:21 gonna move with velocity, the zero constant velocity. Okay, So based
34:26 that, we certainly can predict where ball will be at time point except
34:32 . So at time 10.1, it'll a position acceptance. Here's the predicted
34:38 . Now, of course, Ben . One happens, we say.
34:44 , great. Now I have some , and I expect it to be
34:49 disposition when I concert within that and OK, let me find it.
34:53 , okay. Turns out I found the ball at this location by someone
34:59 this becomes my measurement now, of , given my measurement, I need
35:03 somehow correct. You know what I've and sign to. Thanks. My
35:09 . Extra one and my observation. one right. So which one is
35:14 ? I have to somehow reconciling That an ex wonder. Why one of
35:19 . If if I don't correct, will end up, sort of drifting
35:26 eventually lose track of where the ball . Right? So I hope you
35:31 a way of correcting what this is and the idea of doing this correction
35:36 proposed Ah ah in a free work the mechanism first back in the 19
35:43 by Hari Coleman and they if he apply what he proposed. Uh,
35:51 know, this is confined within the of what we call the Kalman
35:55 It's a It's a simple recursive data Albert that was proposed. And the
36:01 is to generate optimal estimate off the states of the quantities, given a
36:07 off measurements, given a set of right now, optimal in what
36:12 Well, optimal in a linear system which is we're talking about state variables
36:18 can be predicted, uh, within linear framework based on observations.
36:26 of course, it's recursive in the that does it acquitted process. The
36:31 values are updated after every measurement is . OK, and we'll look at
36:36 in the context off our tracking So let's come back to our simple
36:42 problem, right? Tracking a We have all we have the
36:47 eg zero. And we have a V zero. Okay, now,
36:52 course, this is our initialization at zero. And we will also have
36:57 we call a measurement are Sorry. estimation error. Right. So we
37:04 call that e estimate Time 00.0. estimation error. And similarly, we
37:09 . Instead, she ate another quantity the measurement better, which is any
37:15 . Okay, now, EG zero zero are our state variables. So
37:20 gonna happen is that let's say the this time point remind a swan the
37:27 that our ball was at some position e minus one. And it got
37:33 a point, except at time T its velocity v zero and off course
37:43 . In doing this estimation, we an estimation error from nine point B
37:49 one, and this estimation enter. , is, you know, that's
37:55 rough estimate. Time point, t one. Now, remember, our
37:59 model is very simple. Italy, model. It says that X will
38:04 at point. It's all your Pointy. It will be an extra
38:09 , which is nothing but its evolution accepting minus one it's being this position
38:15 the belt rt the time difference Multiplied by the velocity, the
38:22 Now, of course, at time , we're going to search, but
38:26 the vicinity are estimated point. And say my observation tells me that the
38:32 is that Why septic? Now, order to apply Kalman filter to the
38:39 problem that are three steps that beer actually talk about your address. First
38:43 foremost were compute what is called a on gay Kalman. Gain is signified
38:49 or denoted by key sub G, it's equal to the ratio off the
38:57 error at the previous time point divided that estimation error plus ni measurement.
39:05 , now what does really tells us that in general, the value off
39:12 G is going to be between zero one and off course. If the
39:18 estimation error is actually zero, then Kalman gain is same as you.
39:25 police going zero divided by some corn matter to you. On the other
39:30 of the estimation, error is actually greater than our measurement error. Then
39:35 Kalman gain actually approaches close to Okay, the destination error from the
39:41 . Time step is is much larger the measurement error. Then Kalman gain
39:47 be approximately one. Now, once compute the Kalman gain, the really
39:56 for the position estimate for the which is X. Its position at
40:03 Point team is basically reconciling between our at wife's empty and our predicted position
40:14 times that right, which is p t. Which is predicted and
40:20 . Subtitles are measurement. So the consol those is the X predicted a
40:27 point, plus the difference off the position right and that the predicted position
40:36 by the Coleman game. So what this do? It does a couple
40:41 one. Remember if our estimation error the previous step is basically zero
40:47 So, in which case, the Kane is zero. In which
40:50 of course, our predicted point is the same is our actual point at
41:00 , teeth right, which basically says if my estimation is so good,
41:04 I don't need to worry about my being different than my production.
41:11 So my prediction is the correcting. the other hand, if my estimation
41:17 is actually much larger than my measurement . Then Coleman Gina's equals one.
41:24 which case we're saying that well, know, I should really rely on
41:29 measurement rather than my estimation, because estimation error has been large.
41:36 of course, if it's in between two, you come up with some
41:41 off senior combination of the predicted in observed point, and you'll generally lie
41:47 between the predicted and observed points. we might get a correction, which
41:52 us that this might be the actual position off the ball at time
41:58 Okay, you know, of once we have resolved this, the
42:03 step is to compute the new Error on the new estimation error is
42:10 given by one minus the Kalman gain by the previous estimation. So this
42:17 how the updated on estimation error and repeat this process over in order
42:23 So common filter actually is very, simple on at the same time very
42:29 in addressing the tracking problem, So it requires prediction. Step,
42:36 is the previous position, plus dealt multiplied by the velocity, gives us
42:42 predicted position of course, once we that, we actually, once we
42:47 a measurement at Time Point T. , update the necessary elements of Kalman
42:53 , which is the Kalman gain. actual position estimate at nine point
43:02 And of course, the new estimate our in our, uh, time
43:08 . And use these quantities now to ahead and go back to your prediction
43:14 the next time point. And we repeating this process. So this is
43:18 Kalman filter is on bits used extensively simple tracking problems and especially in the
43:27 where the dynamics of the object we're Ah, our senior, this is
43:34 important to keep in mind that the of the object be attracting have to
43:38 linear in the context off applying the Kalman filter. All right, so
43:46 don't we stop here for today? , and we'll continue our discussion next
43:52 in terms of looking at some of other elements that are necessary also.
43:58 , are it's not not that they're , but they're also utilize quite extensively
44:03 Zoe. Think about how we address tracking problem in the context of computer
44:08 applications. All right, but that will stop here for the day.
44:14
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